Role of gravitational decoupling on theoretical insights of relativistic massive compact stars in the mass gap

The detection of gravitational waves by the LIGO Scientific and VIRGO Collaboration has significantly advanced our understanding of classical general relativity (GR), particularly regarding its limitations in modeling supermassive black holes (BHs). Notable events such as GW190814 and GW170817 have provided critical insight into these deficiencies [1, 2]. The GW190814 event highlights a merger between a compact binary system consisting of a BH, estimated to have a mass between 22.2 and 24.3 $M_{\odot}$, and a companion object with a mass ranging from 2.50 to 2.67 $M_{\odot}$. Meanwhile, the GW170817 event, detected on August 17, 2017, is thought to be the merger of two NSs, each with masses between 0.86 and 2.26 $M_{\odot}$. These gravitational wave events are particularly significant in the context of pulsars (PSs), strange stars (SSs), and neutron stars (NSs), which have long served as cosmic laboratories for studying matter at ultrahigh densities. Although Einstein’s classical theory of GR accounts for various observed phenomena, such as compactness, mass-radius relationships, and surface redshifts, it fails to explain the unusual observations associated with NSs that exceed a mass of $M=2M_{\odot}$. This discrepancy underscores the importance of ongoing research into the nature of these compact objects and the fundamental laws of physics that govern them.

Several theories have been proposed to explain the observed signals from gravitational events. These include the characteristics of matter via equation of state (EOS) in the binary star system and various modifications to gravity theories. The signal detected from GW190814 has sparked discussions on the nature of the secondary component, influenced by several factors, including its mass of approximately $2.59^{+0.08}_{-0.09}M_{\odot}$, the absence of significant tidal deformations, and the lack of accompanying electromagnetic signals. This indicates the possibility that the secondary component is an NS or a BH. Consequently, we face a dilemma: it might be the lightest BH or the heaviest NS ever observed in a double compact object system. In the meantime, the GW170817 event and its associated electromagnetic signals offered scientists a valuable opportunity to suggest more unconventional EOSs for the NSs involved in this merger. The electromagnetic radiation associated with GW170817 included two main features: a brief gamma-ray burst, GRB170817A, that was detected approximately 2 seconds after the gravitational wave signal, and a kilonova referred to as AT2017gfo, which peaked in brightness several days after the merger.

Several researchers have been investigating the properties of the binary components based on the observed delays. The kilonova signal indicates a relatively stiff EOS [3, 4]. To explain the potential tidal deformability range of 400 to 800 and the radius of the 1.5 solar mass component, estimated between 11.8 km and 13.1 km – the GW170817 event has been modeled as a merger between a hadronic star (HS) and a SS [5]. Furthermore, the delay in the gamma-ray burst has led some researchers to suggest that the remnant of the merger was probably a hypermassive star that collapsed into a BH in just a few milliseconds [6]. To address the observed small radii and moderate tidal deformability, it has also been proposed that stellar matter undergoes significant phase transitions at supranuclear densities, resulting in the formation of quark matter. In addition, the recently announced GWTC-3 catalogue by the LVC includes new asymmetric systems [7]. Among these, GW200210 stands out as a compact-object merger, showing notable similarities to GW190814. Despite being a relatively low-significance detection (false-alarm rate $>1~{}yr^{-1}$), its discovery suggests that asymmetric systems may be more common than previously thought.

Observations of galactic compact objects suggest a shortage of binaries with masses in the lower mass gap [8, 9, 10, 11, 12], potentially linked to star formation processes, although selection effects may also play a role. The kicks imparted by core collapse supernovae might lead to higher velocities for lower-mass BHs, making them less likely to stay in observable binaries. Compact objects can reach mass within this gap through gas accretion – either from low-mass X-ray binaries or AGN disks – or via mergers, such as NS collisions in dense stellar clusters. However, the expected rates of these mergers are low, especially for the masses observed [13, 14, 15, 16]. BHs in cluster cores tend to dominate, limiting NS mass segregation and leading to a higher rate of BH mergers compared to those involving NSs. Consequently, the likelihood of dynamical origins or AGN-assisted binary mergers involving objects in the lower mass gap appears minimal, as most mergers are expected to be BH-BH events. NS mergers may occur within hierarchical triple systems, where two NSs form from an inner binary and a BH from a tertiary star [16, 17]. The expected rate of such double mergers is about 0.1% to 3% of the binary NS merger rate of 10–1700 $Gpc^{-3}yr^{-1}$, suggesting that some observed NS mergers could lead to secondary mergers [18, 19].

The origins of objects in the lower mass gap can be inferred from their mass. Known Galactic binary NS systems display a narrow mass distribution centered around approximately $2.65\pm 0.12\,M_{\odot}$ [20], suggesting that the remnants of NS mergers likely cluster around a mass similar to about $2.6\,M_{\odot}$ [16]. In contrast, processes such as accretion and stellar evolution may result in a broader mass distribution in this gap, as NSs must accrete several tenths of a solar mass to reach these levels. Even if accretion leads to a narrower distribution, there is no definitive astrophysical reason for it to center around $2.6\,M_{\odot}$. Additionally, if two NSs become gravitationally bound through chance encounters in stellar clusters, they may not originate from a binary system, leading to a wider mass distribution compared to binary NSs [21, 22, 23]. For mass gap objects formed by NS mergers, the BH mass distribution may vary depending on their origin. For instance, GW190814 $(23.2^{+1.1}_{-1.0}\,M_{\odot})$ and GW200210 $(24.1^{+7.5}_{-4.6}\,M_{\odot})$ have similar primary masses. In stellar-cluster scenarios, merger remnants encounter BHs randomly, producing a broader primary-mass distribution. Conversely, hierarchical triple scenarios may lead to a more narrowly distributed primary mass, especially when a heavier tertiary BH stabilizes the system [16], as supernova explosions are less disruptive in these configurations, allowing outer orbits to remain stable.

Our main aim is to simulate the configurations of matter associated with mass gap objects formed by NS mergers and other massive pulsars. While the distribution of BH masses can vary depending on their formation pathways, they often cluster around similar values, such as those seen in GW190814 $(23.2^{+1.1}_{-1.0}\,M_{\odot})$ and GW200210 $(24.1^{+7.5}_{-4.6}\,M_{\odot})$. We shall investigate these phenomena using the gravitational decoupling (GD) approach [24, 25], with additional insights provided in [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] that explore the applications of GD within the framework of standard GR. For further applications in higher dimensions, see [37, 38], for instance. In our examination of the SS model, we place special emphasis on the contributions from the deformation adjusted by the decoupling constant and the special bag function within the bag model. This framework incorporates all necessary adjustments to the energy and pressure functions of the SQM approach, which posits that quarks are noninteracting and massless. By analyzing the observed mass-radius limits of mass-gap objects formed by NS mergers and other massive pulsars, we can effectively constrain the free parameters in our model. This, in turn, enables us to make predictions about the radii and moments of inertia of these objects.

This work is organized as follows: In Sect. 2, we present the gravitationally decoupled Einstein field equations produced by two sources. Sect. 3 explores the minimally deformed solution for SSs, with Subsect. 3.1 addressing the density constraint, $\epsilon(r)=\theta^{0}_{0}(r)$, and Subsect. 3.1 focusing on the pressure constraint, $P_{r}(r)=\theta^{1}_{1}(r)$. In Sect. 4, we discuss the boundary conditions for SS models under gravitational decoupling. In Sect. 5, we will explore the physical viability of our deformed strange star models and their implications in astrophysics. The stability analyses, including those based on the adiabatic index and the Harrison-Zel’dovich-Novikov criteria, are discussed in Sect. 6. Finally, concluding remarks can be found in Sect. 7.

This section provides a focused overview of gravitationally decoupled Einstein field equations for two independent sources

with

where the relativistic units are denoted as $G=c=1$.

The Ricci tensor is designated by the notation $R_{ij}$, where $\mathcal{R}$ represents the contracted Ricci scalars and $\beta$ denotes the decoupling parameter to the field equations. The energy-momentum tensor is denoted as $T_{ij}$, and the source $\theta_{ij}$ has the potential to define other fields, including scalar, vector, and tensor fields. The fact that the Einstein tensor $(G_{ij})$ satisfies the Bianchi identity implies that the effective energy-momentum tensor $T^{\text{eff}}_{ij}$ must be preserved

The following static spherically symmetric line element is being utilized to describe the space-time of the star system’s internal region:

where the metric potentials $\mathcal{F}_{0}$ and $\mathcal{N}_{0}$ are the only ones that are dependent on radial distance.

Taking into consideration the fact that the internal structure of the self-gravitating system, which is characterized by the effective energy-momentum tensor $T^{\text{eff}}_{ij}$, describes an anisotropic matter distribution

where the formula for $u^{i}$ is defined as: $u^{i}=e^{\mathcal{F}_{0}(r)/2}{\delta^{i}}_{4}$, therefore denoting the four-velocity. The unit vector in the radial direction is denoted by the symbol $\chi^{i}$, which is defined as $\chi^{i}=e^{\mathcal{N}_{0}(r)/2}{\delta^{i}}_{1}$. The pressures in the radial and tangential directions are represented by $P^{\text{eff}}_{r}$ and $P^{\text{eff}}_{t}$, respectively, while the energy density of matter is characterized by $\rho^{\text{eff}}$. Furthermore, the $4$-velocity $\chi^{i}$ and the unit space-like vector $u^{i}$ in the radial direction satisfied the conditions: $\chi$. Therefore, the components of the effective energy-momentum tensor $T^{\text{eff}}_{ij}$ can be provided as follows:

Next, the complete formulation of Einstein’s field equations may be expressed as the collection of the following differential equations for the metric (2.1):

Applying the metric function allows for the determination of the mass function, $m(r)$, of a charged perfect fluid sphere

one can get also as equivalent to [39]

By using Eqs. (2.7) and (2.10), we derive

On the assumption that $P^{\text{eff}}_{r}\neq P^{\text{eff}}_{t}$, the condition $P^{\text{eff}}_{r}=P^{\text{eff}}_{t}$ suggests the presence of an isotropic fluid distribution. The anisotropy factor, represented by the symbol $\Delta$, is precisely mathematically defined as: $\Delta^{\text{eff}}=P^{\text{eff}}_{t}-P^{\text{eff}}_{r}$. The term $\frac{2(P^{\text{eff}}_{t}-P^{\text{eff}}_{r})}{r}$ denotes a force that emerges from the anisotropic properties of the fluid. When the pressure $P^{\text{eff}}_{t}$ exceeds the pressure $P^{\text{eff}}_{r}$, the force is directed in an outward direction. When the value of $P^{\text{eff}}_{t}$ is less than $P^{\text{eff}}_{r}$, the force acts with an inward direction. Nevertheless, if the value of $P^{\text{eff}}_{t}$ is greater than the value of $P^{\text{eff}}_{r}$, the force allows for the formation of a more condensed structure under the scenario of an anisotropic fluid as opposed to isotropic fluid distribution [40].

In addition to Eqs. (2.10) and (2.12), the pressure gradient may also be represented in terms of $m$, $q$, $\rho$, and $P_{r}$ using equations (2.7)–(2.9)

which provides the generalized Hydrostatic TOV equation for anisotropic stellar structure [41, 42].

Our next approach is to determine a precise solution for the field equations (2.7)–(2.9) for the TOV equation (2.13) that describes a model of a strange star. We see that the system of field equations exhibits a significant degree of non-linearity, thereby presenting a challenging task in their solution. Hence, we use an alternative method called gravitational decoupling using the minimum geometric deformation (MGD) methodology and a specific transformation associated with the gravitational potential

Let $\mathcal{H}(r)$ and $\mathcal{G}(r)$ represent the decoupling functions, respectively, in connection with the temporal and radial metric components. The deformation may be appropriately tuned by adjusting the decoupling constant $\beta$. When $\beta=0$, the conventional GR-gravity theory is systematically restored. By using the MGD technique, we are able to establish the values of $\mathcal{H}(r)=0$ and $\mathcal{G}(r)\neq 0$. This observation suggests that the appropriate transformation only applies to the radial component of the metric function, while the temporal component remains unaltered. This method of MGD partitions the decoupled system (2.7)–(2.9) into two distinct parts. The initial system is associated with $T_{ij}$, whereas the second system relates to the additional source $\theta_{ij}$. To formulate the initial system, we examine the energy-momentum tensor $T_{ij}$ which characterizes an anisotropic matter distribution presented by

The energy density is denoted by $\epsilon$, while the radial pressure and tangential pressure for the seed solution are denoted by $p_{r}$ and $p_{t}$ correspondingly. Accordingly, the effective quantities may be expressed as

Furthermore, the related effective anisotropy is

The effective anisotropy can be defined as the combined value of two anisotropies in connection to matter distribution namely $T_{ij}$ and $\theta_{ij}$. Gravitational decoupling generates the anisotropy ($\Delta_{\theta}$), which may increase the effective anisotropy. The system (2.7)–(2.9) may be decomposed into two systems by employing the transformations (2.14) and (2.15). The first system is dependent on the gravitational potentials $\mathcal{A}_{0}$ and $\mathcal{B}_{0}$, (viz. when $\beta=0$):

From Eq. (2.13), the following result is obtained:

This is a TOV equation for the system’s configuration (2.19)–(2.21), a solution to which may be found in the spacetime presented below:

By activating $\beta$, one may derive the second set of equations as

The following resulting equation is provided by the linear combination of the Eqs. (2.24)–(2.26) as

The mass distribution for each system may be expressed by the following formula

where the mass functions corresponding to sources $T_{ij}$ and $\theta_{ij}$ are indicated by the variables $m_{GR}(r)$ and $m_{\theta}(r)$, respectively.

Finally, the benefit of MGD-decoupling gets clear: we can generalize any existing solutions that are linked to the matter-spacetime $\{T_{ij},\mathcal{A}_{0},\mathcal{B}_{0}\}$ provided by Eqs. (2.7)–(2.9) and by solving the unconventional gravitational system of equations Eqs. (2.24)–(2.26) to find $\{\theta_{ij}$, $\mathcal{H}$, $\mathcal{G}\}$. Therefore, we may produce the "$\theta$-version" of any $\{T_{ij}\mathcal{A}_{0},\mathcal{B}_{0}\}$-solution as

The above connection outlines a direct approach to investigate the effects of gravity that extend beyond standard Einstein’s gravity.

In order to get a non-singular, monotonically decreasing matter density within the spherically symmetric star system, we choose a modified version of $\epsilon$ as proposed by Mak and Harko [43]:

The constants $\epsilon_{0}$ and $\epsilon_{s}$ represent the central and surface maximum and minimum values of $\epsilon$, respectively. For the sake of studying the SS model, we make the assumption that the distribution of SQM within the peculiar stars is controlled by the straightforward physical MIT bag model EOS [44]. The introduction of the special bag function within the bag model has preserved all the adjustments of the energy and pressure functions of the SQM approach. Our simple bag model assumes that the quarks are both non-interacting and massless. The quark pressure shall be defined as

Let $P^{f}$ represent the individual pressures associated with the quark flavors $(u)$, $(d)$, and $(s)$. These pressures are neutralized by the total external Bag pressure or Bag constant $\mathcal{B}_{g}$. The energy density ($\epsilon$) for the deconfined quarks interior, in relation to the MIT Bag model, can be expressed as following:

Applying Eqs. (3.2) and (3.3) along with the connection $\epsilon^{f}=3P^{f}$, we may express the MIT bag EOS for strange quark stellar objects in its explicit form

At this stage, we have clearly defined the realistic matter density and realistic EOS. Now we focus on the spacetime geometries $\mathcal{A}_{0}$ and $\mathcal{B}_{0}$ for the first system. The differential equation derived from Eqs. (3.1) and (2.19) is expressed as follows:

After doing the integration, we find the potential $\mathcal{B}_{0}$ as follows:

To find the other potential, we combine the EOS (3.4) and Eqs. (2.20) and (3.1), which yields subsequent differential equation

Integrating above Eq. (2.7) that yields a following solution for $\mathcal{A}_{0}(r)$ as

This set of equations (3.6) and (3.8) provides the complete spacetime geometry for the seed solution. Still, given the $\theta$-sector, it is necessary to determine the result of the second set of equations (2.24)–(2.26). Nevertheless, it is important to acknowledge that there are three autonomous equations using four unknown variables. Hence, this scenario necessitates the inclusion of one more piece of information to complete the $\theta$-system, such as $\mathcal{G}(r)$. Considering this problem, we choose the consequent choices:

It can be seen that $\mathcal{G}(r)$ is devoid of any singularities and that $\mathcal{G}(0)$ is equal to zero. Consequently, they enable us to simulate (i) the component $\theta^{0}_{0}$ with the energy density $\epsilon$, namely $\epsilon=\theta^{0}_{0}$ as defined in Eq. (3.9), and (ii) the radial pressure $P_{r}$ with the component $\theta^{1}_{1}$ as defined in Eq. (3.10). This approach has been effectively used by many authors [45, 46, 47, 48, 49] to describe compact objects in GR and alternative gravity theories.

Inspired by those studies, we apply the mimic technique in our current work to solve the system of equations (2.27)–(3.2). Thus, we use the following methodologies $\rho=\theta^{0}_{0}$ and $p_{r}=\theta^{1}_{1}$ to investigate the effects of two density constraints in this study.